Steam Error: Directx9b installed while installing Half-Life 2

This seems to be a problem with not enough disk space on the drive where you are trying to install the game.

There's a check in the Steam installer and a check in the Half-Life 2 installer. The one in the Half-Life 2 installer is failing for some people.

The solution is to free up some disk space on the drive where you are installing the game and run the Half-Life 2 installer again.

You should also make sure to free up enough disk space for future Steam updates or they'll run into problems again in the future.

You must install Steam and HL2 (and all Steam games) on the same disk / partition, you can't install parts on different drives.

Moving Steam Installations: If you need to move your Steam installation to another disk or partition with more room, be sure to create a new desktop shortcut and delete the ClientRegistry.blob file before running Steam.

Also, if you have Steam starting when you run Windows, after moving the installation you need to go and uncheck the Steam option:

Steam > settings > Interface > (uncheck) [ ] Run Steam when Windows Starts

Then restart Steam and go and re-check that option if you still want Steam to autostart.